3d printing with custom surface reflection

ABSTRACT

A method for fabricating custom surface reflectance and spatially-varying bi-directional reflectance distribution functions (BDRFs or svBRDFs). The 3D printing method optimizes micro-geometry to produce a normal distribution function (NDF) that can be printed on surfaces with a 3D printer. Particularly, the method involves optimizing the micro-geometry for a wide range of analytic NDFs and simulating the effective reflectance of the resulting surface. Using the results of the simulation, the appearance of an input svBRDF can be reproduced. To this end, the micro-geometry is optimized in a data-driven fashion and distributed on the surface of the printed object. The methods were demonstrated to allow 3D printing svBRDF on planar samples with current 3D printing technology even with a limited set of printing materials, and the described methods have been shown to be naturally extendable to printing svBRDF on arbitrary shapes or 3D objects.

BACKGROUND

1. Field of the Description

The present description relates, in general, to methods and systems forthree dimensional (3D) printing, and, more particularly, to methods ofcontrolling 3D printing to fabricate an object having user-selectable orcustom surface reflectance on at least portions of the surface of theobject.

2. Relevant Background

Additive manufacturing or 3D printing is a process of making a 3D solidobject, having virtually any shape, from a digital model. 3D printing isachieved using an additive process in which successive layers ofmaterial are laid down or deposited to form different shapes. To performa print, a 3D printer reads the design from a previously stored datafile (e.g., virtual cross sections from a computer aided design (CAD)model or the like) and lays down successive layers of liquid, powder,paper, or sheet material to build the model from a series of crosssections. These layers are joined or automatically fused to create thefinal shape such that 3D printing can produce almost any shape orgeometric feature.

Recent advance in 3D printers have enabled manufacturers and consumersto print 3D objects from 3D models with high accuracy. Printing ofcolored 3D models is also now possible using a multi-material 3Dprinter. It is expected that the popularity and use of 3D printers willcontinue to grow in the coming years especially as costs associated withthe 3D printers continues to decrease even as the accuracy of the 3Dprinters increases over time.

One significant drawback, though, of 3D printing is that the printingdevices typically cannot directly replicate the reflection properties orreflectance of an object being created with a 3D model. For example, itmay be useful to manufacture an object that has two or three differentreflection properties such as being transparent in portions, beinghighly reflective or having a glossy finish in other portions, and/orbeing somewhat reflective (and colored, in some cases) or having a mattefinish in still other portions of the object. However, with present 3Dprinting techniques, the reflection properties of the printed object ormodel are defined by or due to the chosen printing material.

Unfortunately, available printing materials offer a rather restrictiveset of reflectance properties such as matte, slightly glossy (e.g.,polished plastic or the like), and very specular (e.g., transparentplastic). Further, even if one of these materials is available andprovides a desired reflectance for a surface of a printed object ormodel, existing 3D printers only support a small number of concurrentmaterials, which severely limits the gamut of printable reflectance.

SUMMARY

The present description is directed toward a 3D printing system andcorresponding method for 3D printing objects having a custom surfacereflectance. Briefly, the method teaches a technique for printing a 3Dobject with an outer or micro-surface (e.g., a reflective skin or layer)that has one, two, or more sets of reflectance properties.

The reflective skin or layer provides custom surface reflectance byproviding a plurality of reflectance elements (or reflectance-defining3D objects or micro-structures) that each include a body with a base(which may have a circular, oval, or other cross sectional shape such asa triangular or polygonal shape) and an outer surface/sidewall extendingoutward from the base (e.g., each reflectance element is defined by abase shape, a maximum base dimension such as less than about 2millimeters, and a height). The outer surface of each reflectanceelement is used to define reflectance and includes numerous facets oradjoining faces that may be configured (shaped, sized, and so on) sothat they have a normal distribution that approximates an input normaldistribution function (NDF). The reflective skin or outer layer may alsoinclude a diffuse color layer (e.g., the reflective skin may be abi-layered structure) providing with ink providing areas or regions ofone or more colors, and a layer of transparent plastic printed or formedin an additive manner to provide the reflectance elements of themicro-surface to provide or set reflectance properties for each coloredregion of the ink layer (e.g., the reflectance elements may differ overeach colored region to provide differing reflectance).

The inventors recognized from micro-facet theory, which considers asurface as an aggregate of microscopic perfect mirrors with differentorientations, that the fraction of light reflected from the surface in aparticular direction depends on the distribution of the normals of thesemirrors. The resulting reflectance is, therefore, a function of thenormal distribution often represented as a normal distribution function(NDF). In fact, many analytical reflectance models use some variant ofan NDF, which suggested to the inventors that specific reflectances canbe achieved by modifying the micro-scale surface structure of anotherwise highly specular material so as to mimic the perfect mirrors ofthe micro-facet theory.

A significant problem facing the inventors, though, was how to find ordetermine a micro-surface that corresponds to a specific NDF. To thisend, it may be desirable to geometrically model the reflectances so asto encompass both diffuse and specular components, but this presents twochallenges. First, a method is desired to print a combination of diffusecomponents (e.g., colored) and specular components (e.g., transparentplastic). Second, the surface to replicate the specular componentspreferably should be smooth and (locally) convex in order to be suitableto the additive nature of 3D printing.

The first challenge is addressed by printing the micro-surface of thereflective skin using transparent plastic (or other transparent orsubstantially transparent/translucent material) as this allows thereflectance elements to be provided on top of a (mostly) diffuse coloror ink layer, yielding a complete reflectance model. The secondchallenge is addressed using an algorithm (or software routine/program)that creates micro-surfaces made up of individual reflectance elements,which may be dome shaped in some implementations, whose normaldistribution approximates the input NDF. The algorithm may, for example,be based on a Voronoi tessellation of the hemisphere with a discrete setof normals. A least squares optimization may then be used tosuccessfully modify a hemispherical base geometry such that the densityof the facets/faces on the outer surface of the body of each of thereflectance elements matches the input NDF. This ensures the resultingshape is smooth and convex. The specular coefficient is approximated byappropriately scaling the dome or other reflectance element body shapes.For example, the dome shapes may be evenly distributed over 3D modelsaccording to a Poisson Disk distribution, while avoiding strongfeatures. In brief, the inventors describe herein a new computationalmodel for designing micro-structures or reflectance elements toreplicate specular and diffuse reflectances and also describe a model tosimulate and design effective bidirectional reflectance distributionfunctions (BRDFs) with the micro-geometry of each reflective layer orskin provided on a 3D printed object. The inventors are unaware of otherwork to fabricate svBRDF on 3D models.

More particularly, a 3D printed object is provided that may be generatedwith a single printing operation by a multi-material 3D printer or withtwo or more printing operations of 3D printers followed by an assemblystep. The object includes a base layer and a micro-structure layercovering the base layer. The micro-structure layer includes a backsurface (e.g., a planar surface) adjacent the base layer and furtherincludes a front surface opposite the base layer. On the front or outersurface there are provided a plurality of reflectance elements, whichare configured with a geometry so as to define a set of reflectanceproperties (e.g., properties to achieve a user-selected targetreflectance). In some cases, the base layer is formed of materials to bediffuse, and the micro-structure layer is formed of a substantiallytransparent material. Typically, the reflectance elements each are madeup of a body extending outward from the front surface, and the body hasan outer surface with a plurality of facets providing the set ofreflectance properties.

In some embodiments, the facets are optimized so as to be ofsubstantially equal area. In the same or other embodiments, the facetshave geometries based on or corresponding to a normal distributionfunction (NDF) associated with the set of reflectance properties. Insuch cases, the geometry defines a number, a size, and a shape of thefacets. Further, the geometry defines a height and an outer dimension ofa base of the body. For example, but not as a limitation, the base ofthe body may be circular or oval in shape (e.g., to providehemispherical shaped reflectance elements or “domes” that can beconfigured to provide a set of reflectance properties for the 3D printedobject).

In some cases, the object may have two or more target reflectances. Inthese cases, the surface of the micro-structure layer may furtherinclude a second plurality of reflectance elements configured to definea second set of reflectance properties. These properties may differ fromthe set of reflectance properties of the first plurality of reflectanceelements, whereby the 3D printed object has an outer surface with atleast two reflectances (one area can be more specular than another areaon the surface of the object).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a functional block diagram or schematic of a 3D printingsystem of one embodiment showing a controller/computing device providinga module for creating a 3D model with custom surface reflectance and forproviding a 3D model, modified to include a reflective skin or coveringlayer(s) with micro-structures adapted to provide the customreflectance, to a 3D printer;

FIG. 2 is a schematic of a processes or functions performed by thesurface reflectance module as part of controlling reflectance of a 3Dprinter surface of an object;

FIGS. 3A-3D provide a 2D graphic illustrations of construction, as aVoronoi Cell, of a geometry for a micro-structure or reflectance elementto provide an NDF;

FIGS. 4A and 4B illustrate side perspective views of a dome-shapedreflectance element or microstructure before and after optimization,respectively, to equalize sizes/areas of the facets on the outersurface;

FIGS. 5A-5D illustrate top and side views of optimized dome-shapedreflectance elements or micro-structures with increasing specularityand, corresponding, decreasing height;

FIG. 6 illustrates a side view of a 3D printed object printed accordingto the present description; and

FIG. 7 is a graph showing measured effective BRDF on printed samples ofan NDF with exponents 1, 10, 25, 100, 500, and 1000.

DETAILED DESCRIPTION

The present description is directed toward methods and systems for usinga 3D printer to print objects that have custom surface reflectance suchas by providing spatially varying bidirectional reflectance distributionfunctions (BRDFs). For example, one may print a projection screen thatmay be installed with a curved surface, and it may be desirable toprovide differing reflectance properties in differing areas or regionsof the projection screen to as to avoid cross talk of reflected light.This may be achieved through the use of the presently described methodsand systems by providing a diffuse color layer (or ink layer) over whichis printed or provided a micro-structure containing layer formed of atransparent material (such as plastics used with 3D printers). The twolayers may be thought of as a “reflective skin” or “reflectance-definingcoating” for a printed 3D object, and the layer of transparent materialis formed so as to have numerous reflectance elements ormicro-structures each configured to provide a particular reflectance(e.g., with facets providing a micro-surface on the body of thereflectance element that meets a specific normal distribution function(NDF)).

Prior to presenting specifics of the 3D printing systems and methods, itmay be useful to more generally discuss techniques for representing andreproducing appearances of various surfaces or real materials that maybe modeled or mimicked via 3D printing. BRDFs may be used to representthe optical reflection properties of a surface. The micro-facet theorydeveloped by Torrance and Sparrow shows that a surface can beinterpreted as an aggregate of microscopic perfect mirrors of differentorientations. Light is being reflected by these tiny mirrors, and thesurface reflection is guided by the distribution of the mirrordirections (i.e., their normals). A representation of this distributionis a main ingredient in many reflectance models.

Others have proposed a method to fabricate micro-geometry by millingmetal away from an outer surface to yield user-controlled highlight(e.g., specular reflectance only). In this proposed method, the normalsfrom a user-defined NDF were sampled and, then, the method involvesusing simulated annealing to find a CO-continuous micro-surface that canbe tiled and milled. This process shows some interesting results, theachieved reflectance had observable artifacts and the milling processmade the method hard to implement for curved objects. Some in the fieldhave also shown that by overlaying inks with different reflectances, itis possible to print spatially varying BRDFs with a reasonable gamutsuch as from mostly diffuse to low glossy. Others have used paper with astatic micro-geometry and have then selectively printed opaque orpartially opaque ink on portions of that surface to control theresulting specular highlight from specific incident lighting directions.Both techniques only apply to printing on paper and on are not directlyextendable to 3D printers. Further, 3D printers have been used toreproduce material appearance. In particular, it has been shown thatsubsurface scattering can be modeled by 3D printing several layers, andthe results were of a high quality. Unfortunately, though, these 3Dprinter techniques did not apply to 3D printing of spatially varyingreflectance on the surface of objects.

FIG. 1 illustrates a 3D printing system 100 that may be used toimplement the methods of controlling surface reflectance properties of aprinted 3D object as described herein. The system 100 includes a printercontroller (or printer input model generator) 110 that operates togenerate and output a 3D or digital model 150 of an object with areflective skin or cover (e.g., a diffuse/base layer and a coveringtransparent layer with numerous micro-structures or reflectanceelements). The model 150 is provided to a 3D printer 160 that uses printmaterials 162 such as colored inks 164 and transparent materials 166(e.g., plastic or the like) to deposit layers or cross sections (printeroutput 170) of the model 150 to create in an additive manner a 3Dprinted object 171. The 3D printer 160 may be a multi-material printeras shown that can print both layers of the reflective skin/coating ormay be a combination of two or more 3D printers with each being a singlematerial or colored printer (e.g., first print the base or diffuse layer174, second print the micro-structure layer 176, and third apply themicro-structure layer 176 over the base/diffuse layer 174).

The 3D printed object 171 is shown to include a 3D body 172 (which maytake nearly any form and shape such a planar body for use in aprojection screen, a body of a product or prop for a set wherereflection is important, and so on). Over one or more surfaces of thebody 172, a base or diffuse layer 174 is applied such as a layer(s) ofink to provide color for one or more regions of the body surface. Theobject 171 further includes an outer micro-structure layer 176 (e.g., alayer of transparent material such as plastic) with a pattern ofnumerous micro-structures or reflectance elements provided on theoutward facing surface to provide one, two, or more regions with one,two, or more regions with a predefined set of reflectance properties(which may differ from region to region due to the varying geometries ofthe micro-structures).

To create the 3D model 150, the printer controller 110, which may be acomputer or computing device, includes a processor 112 that executescode/program instructions (e.g., in memory) to provide a surfacereflectance module 120 (e.g., to act as a special-purpose computercarrying out the algorithm discussed below). The processor 112 furthermanages input and output devices 114 that allow a user to provide userinput 134 and to initiate the surface reflectance module 120 to providethe 3D model 150.

The processor 112 also manages memory or data storage 130 that is usedto store a variety of data used by module/algorithm 120 to generate adefinition of the geometry of the micro-structures 132. This geometry132 is typically defined based on a particular numerical representationof reflectance such as an NDF and may also be generated based on theparticular printer material 166 used for the transparentmicro-structures and also the inks or other materials 164 used toprovide the base or diffuse layer 174.

To calculate the geometry 132 and then reflective skin of 3D model 150,user input 134 may be processed by the surface reflectance module 120.This input 134 may include a 3D model of an object 135 (e.g., definingthe shape and size of the body 172 of the 3D printed object). Further,the user input 134 may include target reflectances for one or moreregions or areas of the outer surface of the body 172. Again, differentreflectance properties may be set for two or more areas of the outersurface of the body 172 or a single reflectance may be utilized. Theuser input 134 may also include a base shape for the micro-structures(e.g., triangular or another polygon, circular, oval, and so on) or adefault value may be used (e.g., a circular base). Further, the userinput 134 may include the printer materials and/or their opticalproperties 138 for use by the module 120 in determining the geometrydefinition 132 as the geometry may vary with the materials/inks and/orcolors of the base layer 174 of an object 171. The module 120 takes thetarget reflectance(s) 136, the base shape 137, and the printer materials(or corresponding optical properties) and generates the geometries 132of the microstructures to be provided in the micro-structure layer 176of the reflective skin printed on the body 172 of the 3D printed object171 (e.g., with one or more dome-shaped structures configured have anormal distribution approximating an input MDF or BRDF). This mayinvolve retrieving predefined shapes from a database storing a pluralityof the definitions 132 generated with previous preprocessing.

With the general operation of the 3D printing system 100 understood, itmay be useful to discuss operation of the surface reflectance module 120in more detail. To print spatially varying BRDFs, the module 120 may beconfigured to map a given input BRDF to a micro-geometry that can beapplied to the shape of an object (e.g., a micro-structure orreflectance element) and 3D printed on outer surfaces of a 3D printedobject. The module (or described method) allows one to control andincrease the appearance range of 3D printing processes without modifyingthe base materials (printer materials 162). FIG. 2 illustrates functionsor steps of a pipeline carried out by the surface reflectance module 120to control the reflectance of a 3D printed surface. It was assumed thatthe printer 160 can only generate surfaces with a matte and a glossyfinish, which provides two corresponding base BRDFs, f_(b) and f_(s),that are diffuse and specular, respectively. The method carried out bymodule 120 involves replicating the effective reflectance of an inputBRDF model by composing these two base BRDFs with custom micro-geometryof the surface (e.g., a plurality of micro-structures or reflectanceelements each with a geometry providing a large number of facets oradjoining faces to reflect light optimized to have a desired normaldistribution).

In a pre-process 210 (which may or may not be carried out by module 120,which may instead simply use the results provided in database 230), thebased materials 212 of the printer are measured as shown at 213 toobtain the base numerical models or BRDFs, f_(b) and f_(s), as shown at214. Then, the pre-processing 210 includes providing at 216 a set ofnumerical representations of reflectance such as a set of analytic NDFs.A micro-structure may be selected such as a dome-shaped reflectanceelement (e.g., circular base). Given as input the NDFs 216, acorresponding micro-geometry providing a plurality of facets of aparticular shape and number may be computed as shown at 218. At 217, itis indicated that using the analytic NDF 216 as input, the hemisphericalbase shape can be optimized by providing discrete facets such that themicro-structure's geometry 218 has a normal distribution thatcorresponds to the desired NDF 216. This process may involve firstgenerating a sampling of the NDF with blue noise property. For example,such a sampling may be performed to facilitate construction of the dome(or other-shaped reflectance element or micro-structure) by optimizing217 the areas of the facets of a cell of a 3D Voronoi Diagram (asdescribed below). Simulating 220 may be performed to measure 224 foreach of the micro-structures 218 the effect of composing the geometrywith the printing material (as shown with simulation input 215).

Using global illumination, the pre-processing 210 may include simulating220 the resulting BRDF of a surface covered by the reflectance elements(e.g., domes defined at 218) taking into account the measured BRDFs ofthe base material at 213 used by the 3D printer. This analysis allowsthe analytic BRDFs to be fit to the simulated data and, thus, to predictthe effect of both geometry and printing material on the finalappearance of the 3D printed object/model. By optimizing and analyzingdomes or other reflectance elements for different NDFs 216, the space ofreproducible BRDFs is explored with the pre-processing (or model), andthe BRDFs (or associated micro-structures and their geometrydefinitions) are stored in a database 230.

During runtime 240, the surface reflectance module 120 takes as inputdata from the database 230, a target reflectance 250, and the modeledsurface geometry (surface of the transparent layer with a plurality ofmicro-structures) and outputs NDF parameters and density of thestructures/reflectance elements on the surface (with optimization 260)at 270 to provide a surface reflectance that matches the input targetreflectance. The runtime process 240 is a data-driven approach tosynthesize an NDF to match a desired appearance of a printed 3D object.Given as input a target reflectance 250, the runtime processing 240includes determining the geometry of domes (or other-shaped reflectanceelements) at 270 with their placement density adapted on the surface tofind the best approximation of the target BRDF. The inventors validatedthe method carried out by the module 120 by simulating, fabricating, andmeasuring the reflectance of a computed micro-structure surface andgeometry of each structure for various NDFs. Further validation wasprovided through a demonstration that these micro-structures can bedistributed on planar samples as well as on arbitrary 3D models, thusallowing one to print 3D models with spatially varying BRDFs.

The BRDF f(ω_(i), ω₀) of a material can be thought to arise from amicro-surface made up of perfectly specular micro-facets with varyingorientations. The resulting specular BRDF is then governed mainly by thedistribution of the normals of the micro-facets. Intuitively, given alight direction, ω_(i), and a view direction, ω₀, only the mirrors witha normal equal to their half-angle, h, contribute to the reflectedlight. Therefore, the proportion of light reflected is equal to theproportion of micro-surface covered by facets with normal h: f(ω_(i),ω₀) a p(h), where p(h) is the NDF. However, there is no simple directrelationship between an NDF and a BRDF because of the effects of maskingand of the reflection properties of the micro-facets.

With these considerations in mind, a micro-geometry (e.g., a dome withmany facets) is designed that has a controlled NDF as a first steptoward the fabrication of BRDFs. To assist in stating requirements ofthe micro-geometry of each reflectance element, the followinginterpretation of the NDF, p(h) may be stated: given a half vector, h,with which one can observe the surface, p(h)dΩ is the probability that arandomly picked micro-facet of the surface lies in the infinitesimalsolid angle dΩ around h. This requirement is continuous and realizing asurface from a continuous NDF is related to normal field integration,which is not always possible and depends on boundary conditions. Also,in the case of an NDF, there are no constraints on the spatialrepartition of the normals so that the translation of an NDF into anormal field is also often not straightforward.

Hence, the inventors determined it would be better to construct asurface from a discrete set of normals that reconstruct the input NDF.Hence, given an NDF, a surface is generated with a discrete number ofnormals that fulfill the following guidelines: (1) the surface shouldhave one facet for each normal in the sampling; (2) all facets shouldhave roughly the same area; and (3) the sampling should reconstruct theinput NDF.

FIGS. 3A to 3D illustrate NDF micro-geometry construction using aVoronoi Cell. Given a sampling of an NDF, n_(i), as shown in graph 310of FIG. 3A, a 3D Voronoi diagram 330 can be constructed (as shown inFIG. 3C) of the points p_(i)=r_(i)n_(i) and the origin (see graph 320 ofFIG. 3B), where the distance r_(i) are positive numbers. The restrictionof the Voronoi diagram to the central cell as shown in FIG. 3D is aconvex surface having exactly one face for each normal of the sampling.FIGS. 3A to 3D are a 2D illustration of the construction of the NDFmicro-geometry as a Voronoi Cell. From a set of normals n_(i) (see FIG.3A), the NDF geometry (see geometry 340 in FIG. 3D) is generated as aVoronoi Cell (see graph 330 of FIG. 3C). To construct the Voronoidiagram, a point, p_(i), is created along each normal of the sampling320 shown in FIG. 3B.

To adapt the areas of the faces of the dome, the distances of the pointsto the origins are interpreted as weights. To each normal vector, n_(i),a radius, T_(i), is assigned, and the distances of the point along thenormal, n_(i), are assigned to the origin. The observation can then bemade that decreasing the distance, r_(i), increases the area, a_(i), ofthe face with normal, n_(i), and, conversely, increasing the distance,r_(i), decreases the area, a_(i).

To generate a discrete dome (or other shaped reflectance element) thatrepresents a desired NDF, the sampling of the normals preferablyapproximates as accurately as possible (or practical) the NDF. Moreover,as discussed next, the areas, a_(i), should be optimized in order toreconstruct the NDF. Further, to be able to optimize the areas so thatall faces have approximately the same area, it is helpful to have theinitial normal sampling have the blue noise property. The ability toadapt the area of the facets depends on the freedom to have the facesgrow or shrink. Hence, it is preferred that the distance betweenneighboring samples be maximal. To this end, one good sampling strategyto fulfill these two goals is to use a method based on Lloyd'srelaxation. For example, one may employ Capacity-Constrained VoronoiTessellation, which is an improvement of Lloyd's method that ensures theblue noise property as well as the adaptation to a distribution. Inorder to use this sampling technique, the orthogonal projection of thehemisphere to the plane may be used.

Next, it may be useful to explain in more detail how optimization of theface areas may be performed within the 3D printing system andcorresponding method(s). Provided an adapted sampling of an NDF,building a reflectance element (e.g., a dome-shaped micro-structure)with equal distances, r_(i), from the points to the origin yields asurface whose areas/facets are highly unbalanced. This can be seen fromthe dome-shaped reflectance element or micro-structure 410 of FIG. 4A (adome before optimization), which has a body 412 defined by an outersurface 414 that extends out from a circular base 416 and the surface414 has plurality of facets/faces 418 with unequal areas.

However, it is preferred that the micro-structure (which typically isformed of a transparent plastic or similar printing material used with3D printers) have facets with similar or substantially equal areas. Sucha dome-shaped reflectance element or micro-structure 420 is shown inFIG. 4B (a dome after optimization), which has a body 422 defined by anouter surface 424 that extends out from a circular base 426 and thesurface 424 has a plurality of facets/faces 428 with equal or at leastsimilarly sized areas. Both domes 410, 420 have the same normals but theareas are balanced after optimization. The inventors further evaluatedthe area optimization algorithm implemented to show typical convergenceof the algorithm, to inspect the distribution of the areas of the facetswith respect to the mean, and to check the algorithm using a plot of theweighted histogram of the areas of the facets. This verified that theresulting or optimized domes (e.g., micro-structure or reflectanceelement 420 of FIG. 4B) reconstructed the input NDF. It was observedthat while the distributions with low exponents are reconstructed veryprecisely, reconstructing the peaks of very specular distributions canbe more challenging.

Optimizing may be performed to balance the areas of the faces by leastsquare optimization. Since it is desired that the areas, a_(i), beequal, the error can be computed with respect to the mean, M, of theareas using the following equation:

$E = {\sum\limits_{i}\left( {a_{i} - M} \right)^{2}}$

To solve the optimization problem, an iterative optimization procedurecan be run. The areas are corrected (made similar/equal) by changing thedistances in a direction that decreases the error on the faces/facetsareas. More precisely, at each iteration, the distance, ri, are updatedproportionally to the distance of a_(i) to M as shown by:

For all i,r _(i) ←r _(i)−α(a _(i) −M)

The iteration is continued until the error, E, is stable. By choosing ato be small enough, the algorithm converges to a minimum. To improve thestability, the vector, r=(r_(i)), is scaled so that its largestcomponent is equal to one. The number of iterations used to reach asatisfying minimum depends on the distribution and on the number ofsamples.

At this point of the description, it may be useful to discuss theeffective BRDF of the micro-geometry (e.g., the geometry defining themicro-structure such as a dome as well as the number, shape, and size ofthe facets/faces on its outer surface). Even though a micro-geometry isgenerated with a precise normal distribution function, the resultingBRDF is not known a priori. In this regard, the micro-facet theory doesnot propose a one-to-one mapping between NDF and BRDF. The BRDF alsodepends on the arrangement of the facets, which influences the geometryattenuation factor. In addition, the printing material is not perfectlyspecular and, therefore, the composition of NDF and base BRDF should beconsidered.

Models for geometry attenuation factors have been developed by othersbut usually only consider V-groove geometry. This is because shadowingand masking is a global phenomenon and it is, therefore, difficult toderive analytic models for this phenomenon. Models that take intoaccount the BRDF of the base material have also been developed. Thecomposition of a BRDF has been modeled with a normal map by aconvolution. For example, this model was used in one prior applicationand a deconvolution was applied to the NDF by the BRDF of the milledmaterial to compensate for the blurring effect of the base material.More recently, bi-scale materials were edited and rendered consideringboth approaches. Starting from the rendering equation, these approachesexpressed the effective BRDF of a surface patch as the integral of theproduct of the base material and a BVNDF (bidirectional visible NDF).The model is similar to convolution of a BRDF and an NDF, but the BVNDFtakes shadowing and masking into account as well. In practice,evaluating the BVNDF is done by rendering the geometry under differentlighting and viewing directions. This method provided to be suitable toaccurately and efficiently render bi-scale materials but does not allowone to produce an NDF from a target BRDF.

To produce an NDF from a target BRDF, the inventors chose, in oneimplementation of the 3D printing system and method, to use rendering tomeasure effective reflectance of the micro-geometry of a reflectanceelement designed to be provided on a printed 3D object. In the followingdiscussion, the assumptions and printing constraints are reviewed thatwere specific to a dome-shaped implementation of a reflectance elementor micro-structure. The description then provides a model for theeffective reflectance of the printed material. This includes a showingof how, using renderings, one is able to predict the effect of themicro-geometry taking into account visibility was well as base BRDFs.Then, a database of BRDFs is built that can be used to reproduce atarget reflectance.

In one prototype implementation, the 3D printer utilized in the 3Dprinting system (and to carry out the 3D printing methods) was an Objet™Connex 350 (distributed by Stratasys Ltd.) in part due to its geometricaccuracy and its ability to print glossy materials. It is desirable toprint or deposit a very specular material to be able to reproduce thewidest range of specularity. In the following discussion, it is assumedthat one can print the micro-geometry of a reflectance element (e.g., adome with many facets on its outer surface) with a specular BRDF, f_(s).

A constraint on the use of domes with circular bases is that the entireouter surface of a dome's body cannot be covered entirely. In the caseof isotropic NDFs, the domes have a circular boundary (or base) so thatthe maximum density the can be achieved is the density of a circlepacking. The remaining surface or background of the dome's bodycontributes to the effective reflectance and, since the roughness ofthis surface was not controlled in the prototype implementations, it istypically desirable to have an underlying base layer formed of materialthat is as diffuse as possible (with the base layer or its materiallabeled as background material, f_(b)).

With regard to representation of the printer materials, an analyticmodel may be used to model the printer materials, with the model beingfit to actual measurements. In one implementation of the 3D printingsystem/method, TangoBlack+ covered with support material was used, whichhas a matte appearance, for the diffuse material, and VeroBlack+ withoutsupport material was used for the specular material. Thin layers ofthese materials were printed and then bent onto a cylinder to be able tocapture a large range of incident light directions and view directionsin a single shot.

FIGS. 5A-5D show top and side views of exemplary dome-shaped reflectanceelements or micro-structures 510, 520, 530, 540 with increasingspecularity as shown with increasingly large specular regions 519, 529,539, 549. Each dome 510, 520, 530, 540 is defined by a circular base516, 526, 536 having an identical diameter, D_(Base) (e.g., 1 to 2 mm orless in some preferred cases). The structures 510, 520, 530, 540 havebodies 512, 522, 532, 542 defined by outer surface or sidewall 514, 524,534, 544 extending outward from the outer periphery of the base 516,526, 536, 546, and the height, H₁ to H₄, of the bodies 512, 522, 532,542 defines the structures and also is shown to decrease with increasedspecularity (as seen from the top views with noiuial incident lightwhere the proportion of the surface reflecting light towards the viewerincreases), e.g., the domes become flatter to provide increasedspecularity. Each surface 514, 524, 534, 544 includes a plurality (e.g.,50 to 100 or more depending on the OD of the domes and the resolution ofthe 3D printer) of facets/faces 518, 528, 538, 548 that have beenoptimized (e.g., to have substantially similar (or even the equal) area)such as with the Blinn-Phong distribution or other technique. In otherwords, a modified Blinn-Phong BRDF model was fit to the measuredsamples, and the inventors' experiments showed that using a model withtwo Blinn-Phong specular lobes results in good fits of the measurements.

With regard to providing a model for the effective BRDF, models for theeffective reflectance of rough surfaces have been used and have provento be suitable for rendering. One common approach provides a formulationthat sums the infinitesimal contribution of all micro-facets to theglobal reflection and relies on a probability distribution. Otherformulations, though, start from the rendering equation, which may bebetter suited to the present application since it allows one to takeinto account the materials of the surface as well as the geometry.

Considering a surface with macroscopic normal, n, covered with domeshaving faces with normals, n_(j), and associated areas, a_(j). Thesurface is printed with the BRDF, f_(b), and a proportion, α, of thesurface is covered by domes printed with the BRDF, f_(s). The effectiveBRDF, f, of such a surface can be written by the following equation,when neglecting the visibility factors:

$\begin{matrix}{{\overset{\_}{f}\left( {\omega_{o},\omega_{i}} \right)} = {{\left( {1 - \alpha} \right){{fb}\left( {\omega_{i},\omega_{o}} \right)}} + {\frac{\alpha}{{\overset{\_}{n} \cdot \omega_{i}}{\overset{\_}{n} \cdot \omega_{o}}}\frac{\sum_{j}{{f_{s}\left( {n_{j},\omega_{i},\omega_{o},} \right)}{a_{j}\left( {n_{j} \cdot \omega_{i}} \right)}\left( {n_{j} \cdot \omega_{o}} \right)}}{\sum_{j}{a_{j}\left( {n_{j} \cdot \overset{\_}{n}} \right)}}}}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

This model shows that the reflectance of the printed surface can bemodulated by varying α. The other degree of freedom is the normaldistribution of the domes. Although this model may not be practical foroptimizing the NDF geometry, it shows that under some approximations theeffective BRDF of the surface can be written:

f (ω₀,ω_(i))=(1−α)f _(b)(ω_(i),ω₀)+αf _(micro)(ω_(i),ω₀)  Equation 2

With this in mind, the following discussion presents use of this modelto simulate the effective BRDF with an outer printed surface havingnumerous micro-structures in the form of many-faceted domes.

To verify that the use of the reflectance elements described hereinprovides control over the reflectance of a surface of 3D printed object,the measurements for planar samples was simulated. In particular,dome-shaped reflectance elements or micro-structures were generated fora Blinn-Phong normal distribution, p(h)=(m+2/2π)(h·n)^(m), with theexponent, m, in the range of [1,1500]. To maximize the density of thedomes on the surface, the domes can be distributed on a planar samplewith a hexagonal tiling. The images are rendered for a high number ofview directions (e.g., 500 or the like) and with normal light incidence.

For each image, a sample of f is extracted by summing the radiance onthe planar sample and dividing by the projected area of the square onthe image. The simulation may be performed using the model fitted to thereflectance of the printer materials. The samples may be measuredwithout rendering the background but only domes distributed upon it.This allows the measurement of the effective BRDF, f_(micro), of thedomes. Even though this does not specifically correspond with theprinted samples, this experiment allowed the inventors to more directlyevaluate the quality of the generated reflective skin/covering with atransparent layer having numerous micro-structures in the form ofmulti-faceted domes.

A set of planar samples were manufactured to implement and validate theideas presented herein. Specifically, to fabricate planar samples aswell as samples for measurements, an Objet Connex350 printer was usedfor the 3D printing of a 3D object. The printer allowed a material to beprinted with a glossy finish or a matte finish. The matte finish wasachieved by depositing or printing a layer of support material on top ofthe printed surface. The support material was then removed, and theportion of the surface that was in contact with the removed material hada matte finish. This is useful because it was desirable to print thesurface between the domes with a glossy finish, but this also representsa limitation for printing arbitrary surfaces. Typically, only high fieldsurfaces can be printed with a glossy finish. The domes in oneimplemented experiment were each composed of 1000 facets on the outersurface of the hemispherical-shaped body. On the printed models, it maynot be readily possible to distinguish the individual facets, but alarge number of facets is desired to run the simulation since domes withfewer facets may result in noisier effective BRDFs (however, the numberof facets may vary widely from up to 100 to 1000 or more depending onthe desired results, printer resolution, and other factors). A minimalsize that may be useful for the domes/reflectance elements has not beendetermined by testing but, by experience, it is believed by theinventors that domes with a radius of up to 3 mm or larger (base havinga maximum outer dimension of up to 6 mm or larger) may providesatisfying results while some implementations may utilized domes withradii of 1 to 3 mm while others may use a radii of less than about 1 mm.

With regard to printing color examples, the above discussion involvedthe use of a diffuse material and a specular material of the same color.The described methods and systems are also applicable to color printing(differing color used for the diffuse/colored layer underneath thespecular material layer with the reflectance elements). For example, theinventors worked with gray scale BRDFs and with black materials. For thetechnique to be useable in a broader range of applications, one mayutilize color diffuse material. This problem may be challenging withcurrent printing technologies but will likely prove less troublesome inthe near future with enhanced printers and printing materials.Generally, 3D printers that are able to print full color (such as theZPrinter) are presently available, but the finish of the surface can berough as a result of the printing process based on powder. The printoutsare initially very matte, and one may cover the printouts with glue,wax, or epoxy to get a glossy finish. However, this process affects thewhole surface, and it seems very hard to cover only thedomes/reflectance elements.

To leverage this limitation, it may be useful to print transparent domes(or other-shaped reflectance elements) and mate them with a colorsurface. This may be done with a single 3D printer or the dome layer maybe printed and then attached onto a color surface layer printedseparately. For example, using a clear transparent material such asVeroClear™, highly glossy transparent domes may be printed with a 3Dprinter. The primary reflection of the dome contributes to the specularreflection, and the light that is transmitted into the micro-geometry isreflected by the diffuse, color material, which adds one or more colordiffuse components to the global reflection of the surface of the 3Dobject.

Although in this implementation, other effects such as Fresnelreflection and caustics cast on the base surface should be taken intoaccount, the inventors believe that this strategy may be useful such aswhen the printing technology allows printing of a clear transparentgeometry over a diffuse surface. This concept was demonstrated as viableby producing a 3D printed object 600 that is shown in side view in FIG.6. The 3D printed object 600 includes a body 610 (e.g., a cow-shapedbody) with an outer colored diffuse layer having areas with differingcolors 612 and 614, for example. The object 600 further includes anouter micro-structure layer 620 with a plurality of micro-structures orreflectance elements 622 (e.g., dome-shaped protrusions extending outfrom a planar surface/layer). The micro-structure layer 620 may have afirst region (such as near the nose of the cow to cover region 612) thatis more specular while a second region (such as over black spots 614) ismore diffuse (i.e., the micro-structures may have two configurations(shapes and/or heights) to provide two custom reflectances or two setsof reflectance properties on the 3D printed object.

The 3D object 600 was used by the inventors to demonstrate that theprinting techniques described herein are viable for producing a printedexample with varying diffuse color and specularity. To produce thesample object 600, a diffuse model was printed using a ZPrinter andinsets on the surface were created where transparent domes wereplaced/attached that were printed separately using VeroClear™ with anObjet™ Connect350 printer. For the 3D model, a constant density of thedomes 622 was used over the diffuse surface 610 of the printed model.The distribution of the domes 622 was done using the Poisson Pointsampling algorithm discussed above. The surface of the transparent layer620 was covered with micro-structures having a geometry optimized forthe modified Blinn-Phong NDF with exponents 10, 50, and 100. In thisparticular example, the nose region 612 of the cow's body 610 wascovered with the most specular domes 622, and two other NDFs were usedto increase the contrast between the fleck or spot regions 614 and therest of the cow's body 610.

To validate the model further, Equation 2 may be used to compare theappearance of the theoretical and actual effective BRDF, f. The domesmay be distributed, for example, on a sphere using the vertices of anicosahedral tessellation as the location of the domes (or other-shapedreflectance elements). With this tessellation, the density of thesurface covered by the micro-geometry is equal to α=0.86. The referenceimages may be rendered using Equation 2 and with f_(micro) in theachievable range. The fitted model of the printing materials for f_(b)was used for both rendering and f_(s) for the base BRDF of the domes. Arendering of such a sphere under environment lighting with a range oftarget reflectances was performed and showed the reconstructedreflectance using the inventors' model compared well with a targetreflectance (target sphere).

To measure the BRDF of the base materials and of the domes with agenerated micro-geometry, an image-based approach was used. A cylinderwas printed that was covered with domes having a micro-geometry taughtherein (using an NDF(s)), and the BRDF was measured on a row while thecylinder was rotating around its axis of symmetry. This setup allowedthe inventors to perform the integration over a surface patch involvedin the model of the effective BRDF. By taking long exposures of therotating sample, integration can be performed in one direction. Theother integration was done by averaging vertically the radiance ofseveral rows having the same dome design. The measurement set upincluded a turntable on which the sample was placed. The set up alsoincluded a standard digital camera and a rectilinear light source. Themeasurement set up was surrounded by black cloth to avoid lighting fromthe environment. A rectilinear light source was used to allow theassumption that the incoming light directions were in a horizontalplane. The light source was a thin neon tube chosen to provide a whitespectrum. HDR images were taken of the samples with the scaling of thecurves being recovered by measuring the radiance reflected by a diffuseSpectralon sample under the same illumination conditions.

As part of this prototyping or validation process, the samples of thedatabase (exponents 1, 10, 25, 100, 500, and 1000) were printed as to bedistributed on a hexagonal lattice and with maximal radii. Graph 700 ofFIG. 7 illustrates effective BRDFs measured on printed samples forexponents of the NDF 1, 10, 25, 100, 500, and 1000. It was verified thatthe measured BRDF have a profile resembling the Blinn-Phong BRDF. Theexponents and specular coefficient of the measured material are higherthan the ones observed in simulation. This difference may mainly becaused by the printing resolution. The 3D printer that was used printsthe geometry/micro-structure layer in a layer-by-layer manner and eventhough these layers are very thin, the printed domes are not smooth butstair-shaped. As a result, the proportion of the surface being flatincreases, which results in a higher specularity.

With regard to results achieved, planar samples were first generatedwith spatially varying reflectance. Gray level samples were initiallyprinted since the pipeline was run with black printer materials. Thesesamples were fabricated using an Objet™ Connex350 3D printer. Both thebackground surface (or diffuse layer) and the micro-structure layer wereprinted with VeroBlack+ in one run with the background printed with amatte finish. To demonstrate that the described micro-geometry can alsobe applied on 3D models, a color model was printed (e.g., the body ofthe cow shown in FIG. 6) with a ZPrinter. Then, this model was coveredwith a layer of transparent material (e.g., VeroClear™) with 3D domesprinted with an Objet™ Connex350 3D printer.

The 3D printing methods (or algorithm) taught herein can also handleanisotropic BRDFs and optimize domes (or other-shaped reflectanceelements) for the anisotropic Phong normal distribution. An anisotropicdistribution may be generated and then the corresponding geometry of thedomes of the transparent layer can be computed. It will be understoodthat specularity increases with an increase in a “vertical” direction(e.g., a circular dome will be less specular than a more oval dome andthe more oval or elongated the dome the greater the specularity of thedome-shaped reflectance element). The normal distribution that wasreconstructed is given by:

$\begin{matrix}{{P_{h}(h)} = {\frac{\sqrt{\left( {n_{u} + 1} \right)\left( {n_{v} + 1} \right)}}{2\pi}\left( {h \cdot n} \right)^{{n_{u}{\cos^{2}{(\varphi)}}} + {n_{v}{\sin^{2}{(\varphi)}}}}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

where φ is the angle between the projection of the half vector and thefirst axis of the tangent plane. Renderings show that a circulararrangement of the domes/structures on a micro-structure layer producesa brushed metal aspect. Disk-shaped micro-structure layers were renderedwith anisotropic domes aligned along concentric circles for differentcombinations of the two NDF parameters, n_(u) and n_(v). Otherarrangements of the domes/reflectance elements may be used to provide adesired density of the micro-structures (e.g., may be desirable to havean increased density of reflectance elements on a surface of themicro-structure layer). Printed disks with such anisotropic NDF geometrywere produced and inspection was performed under the illumination of aspherical light source. The light direction varied from normal incidenceto a direction almost aligned with the view direction.

Although the invention has been described and illustrated with a certaindegree of particularity, it is understood that the present disclosurehas been made only by way of example, and that numerous changes in thecombination and arrangement of parts can be resorted to by those skilledin the art without departing from the spirit and scope of the invention,as hereinafter claimed.

The description presents a method for 3D printing spatially varyingBRDFs. At the core of the method is a computational model forefficiently designing printable micro-structures that replicate a givenNDF. To replicate an input BRDF, the description provides a data-drivenapproach that models the effective BRDF as two components, i.e., adiffuse background material layer and a micro-structure layer (e.g., atransparent layer with specular dome-shaped reflectance elements withdesired, custom facet distributions). This design choice was motivatedby available multi-material 3D printers that provide materials withvarying specularity. For 3D printing, prototypes of 3D objects wereprinted using an Objet™ Connex350 machine, which can only reproduceglossy surfaces if they are not in contact with support material. Thiscurrently limits one-pass fabrication to geometries that can berepresented as a height field. More complex shapes were fabricated,though, using a two-step process, which involved an assembly of thediffuse object shape and the separately printed micro-structurelayer/film. However, given the rapid development of additivemanufacturing and available base materials, it is likely that thiscurrent limitation will soon be resolved as 3D printers become moretechnologically advanced.

For analyzing and compactly representing the effective BRDF of thedescribed micro-structures, the Blinn-Phong model was used as this isone of the most widely used analytic models in computer graphics.However, there are many other analytic models that could provide useful.Further, it may be desirable to investigate and characterize the gamutof reproducible BRDFs with the micro-structures described herein. In theabove description, a hemispherical base shape was used in themicro-geometry (in the micro-structure layer), and this shape wasoptimized such that its normal distribution corresponded to a desiredNDF. However, many other shapes are believed to be well suited for usewith the described techniques. Still further, the density of themicro-structures may be optimized to suit a desired set of reflectanceproperties. With this in mind, it is recognized that, similar to asphere-packing problem, the circular footprint of the dome-shapedreflectance elements limits the percentage of the surface that can becovered with this base shape. Hence, embodiments of 3D printed objectsmay include a micro-structure layer with reflectance elements with otherbase shapes that may be relatively easy to manufacture and apply onarbitrary 3D objects.

We claim:
 1. A 3D printed object, comprising: a base layer; and amicro-structure layer covering the base layer, wherein themicro-structure layer comprises a back surface adjacent the base layerand a front surface opposite the base layer, wherein the front surfacecomprises a plurality of reflectance elements, and wherein thereflectance elements are configured to define a set of reflectanceproperties.
 2. The object of claim 1, wherein the base layer is formedof materials to be diffuse and wherein the micro-structure layer isformed of a substantially transparent material.
 3. The object of claim1, wherein the reflectance elements each comprise a body extendingoutward and wherein the body comprises an outer surface with a pluralityof facets providing the set of reflectance properties.
 4. The object ofclaim 3, wherein the facets are of substantially equal area.
 5. Theobject of claim 3, wherein the facets have a geometry based on a normaldistribution function (NDF) associated with the set of reflectanceproperties.
 6. The object of claim 5, wherein the geometry defines anumber, a size, and a shape of the facets.
 7. The object of claim 6,wherein the geometry defines a height and an outer dimension of a baseof the body.
 8. The object of claim 7, wherein the base of the body iscircular or oval in shape.
 9. The object of claim 1, wherein the surfacefurther comprises a second plurality of reflectance elements configuredto define a second set of reflectance properties differing from the setof reflectance properties of the plurality of reflectance elements,whereby the 3D printed object has an outer surface with at least tworeflectances.
 10. A 3D printing method, comprising: providing a diffusebase layer; printing a layer of transparent material comprising asurface with a plurality of micro-structures, wherein themicro-structures are configured based on a 3D model defining a geometrycorresponding to a numerical representation of a target reflectance; andforming a 3D printed object by attaching the layer of transparentmaterial to the diffuse base layer with the micro-structures facing awayfrom the diffuse base layer to define reflectance for the 3D printedobject.
 11. The method of claim 10, wherein the providing, the printing,and the forming are performed by a single multi-material 3D printer. 12.The method of claim 10, further comprising measuring reflectances ofprinting materials available for the diffuse base layer and the layer oftransparent material and, based on the measured reflectances, optimizingthe geometry of the micro-structures to reproduce the numericalrepresentation.
 13. The method of claim 10, wherein the numericalrepresentation comprises a normal distribution function (NDF).
 14. Themethod of claim 13, wherein the geometry of each of the micro-structuresis provided by performing a Voronoi optimization.
 15. The method ofclaim 14, wherein each of the micro-structures comprises a base with oneor more sidewalls extending outward from the base and wherein outersurfaces of the one or more sidewalls comprises a plurality of facets.16. The method of claim 15, wherein the facets each have substantiallyequal areas.
 17. The method of claim 15, wherein the base is circular oroval in shape, whereby the body is dome shaped.
 18. A method for used in3D printing an object with customizable reflectances, comprising: for aset of printing materials for a 3D printer, determining a bidirectionalreflectance distribution function (BRDF) for a base diffuse layer andfor a specular outer layer; with a set of normal distribution functions(NDFs), computing a corresponding geometry for a multi-facetedmicro-structure; measuring reflectances of the specular outer layer witha surface of the specular outer layer comprising a plurality of themulti-faceted micro-structures corresponding to each of the NDFs withthe set of printing materials; simulating a resulting BRDF of the basediffuse layer covered with the specular outer layer with themulti-faceted micro-structures; with an input target reflectance,determining a geometry of the multi-faceted micro-structures and adensity on the surface of the specular outer layer to synthesize one ofthe NDFs; and operating a 3D printer to print a 3D object using thedetermined geometry and the set of printer materials.
 19. The method ofclaim 18, wherein the determining the geometry comprises performingVoronoi optimization of to define facets of the micro-structures. 20.The method of claim 18, further comprising with an additional inputtarget reflectance, determining an additional geometry of themulti-faceted micro-structures to synthesize another one of the NDFs andwherein the operating of the 3D printer comprises printing the 3D objectusing the determined geometry, whereby the printed 3D object has two ormore reflectances.